Distributionally Robust Goal-Reaching Optimization in the Presence of Background Risk

Abstract

In this article, we examine the effect of background risk on portfolio selection and optimal reinsurance design under the criterion of maximizing the probability of reaching a goal. Following the literature, we adopt dependence uncertainty to model the dependence ambiguity between financial risk (or insurable risk) and background risk. Because the goal-reaching objective function is nonconcave, these two problems bring highly unconventional and challenging issues for which classical optimization techniques often fail. Using a quantile formulation method, we derive the optimal solutions explicitly. The results show that the presence of background risk does not alter the shape of the solution but instead changes the parameter value of the solution. Finally, numerical examples are given to illustrate the results and verify the robustness of our solutions.

ACKNOWLEDGMENTS

The authors are grateful to an editor and two anonymous referees for their helpful suggestions that greatly improve the quality of the aritcle.

FUNDING

Y. Chi was supported by grants from the National Natural Science Foundation of China (Grant No. 11971505) and 111 Project of China (No. B17050). Z. Q. Xu was partially supported by grants from NSFC (No. 11971409), Hong Kong GRF (No. 15202817 and No. 15202421), the PolyU-SDU Joint Research Center on Financial Mathematics and the CAS AMSS-POLYU Joint Laboratory of Applied Mathematics.

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Notes

1 Embrechts et al. (2014) gave a very detailed discussion on why estimating the dependence structure between risks is statistically and computationally challenging.

2 Wüthrich (2003) emphasized that usually in actuarial problems it is difficult to have a good intuitive feeling for the dependence structure and one has not enough data to really analyze the dependence structure. McNeil, Frey, and Embrechts (2015) claimed that, in the foreseeable future, the lack of operational loss data is a big concern in modeling the dependence structure.

3 Notably, the robust (or worst-case scenario) portfolio selection and optimal (re)insurance design have been explored in the literature. For example, Balbás, Balbás, and Balbás (2016) considered a robust portfolio selection problem by minimizing the risk for a given guaranteed expected return. Hou and Xu (2016) analyzed a distributionally robust portfolio selection problem under a mean-variance framework instead of goal-reaching. Asimit et al. (2017) investigated the optimal insurance contract with model risk in the robust optimization sense.

4 It should be emphasized that sometimes we have to extend the domain of $F_X^{-1}(t)$ to $(0,\infty )$ by setting $F_X^{-1}(t)=+\infty$ for t > 1.

5 Throughout the article, the terms “increasing” and “decreasing” mean “nondecreasing” and “nonincreasing,” respectively.

6 The martingale approach can be labelled as a two-step scheme. It first identifies the optimal payoff $X^{*}$ by solving Problem (2.5) and then derives the optimal portfolio $\pi (·)$ by replicating the optimal final payoff $X^{*},$ where the theory of backward SDE is applied. See Bielecki et al. (2005) for more details on this approach. Because the second step is rather standard, we focus only on the first step in this article.

7 Xu, Zhou, and Zhuang (2019) provided a detailed discussion on the incentive-compatible condition. It is worth noting that the value change of $I\prime (x)$ on a set with zero Lebesgue measure has no impact on I(x). Therefore, we do not repeatedly emphasize the term “almost everywhere” when mentioning the marginal ceded loss function $I\prime (x).$

8 It is worth mentioning that the worst-case dependence structure between the insurable risk X and the background risk Y that solves the optimization problem on the left-hand side of Equation (3.13)(3.13) $$\underset{\Xi \sim F_{\pi }}{\inf }\mathbb{P}\left(R(X)⩽\Xi \right)=\underset{z\in \mathbb{R}}{\sup }(F_{R(X)}(z)-F_{\pi }(z)).$$(3.13) is not (completely) comonotonic. In fact, the results in Appendix B indicate that the worst-case dependence structure between X and Y is associated with a so-called shuffle of min. Such a family of dependence structures shows that X and Y are strongly piecewise monotone functions of each other or piecewise comonotonic (see, e.g., Embrechts, Höing, and Puccetti 2005).

9 Bahnemann (2015), an actuarial report from the Casualty Actuarial Society, pointed out that one of the most popular probability distributions used to model the size of insurance claims is the truncated and shifted Pareto distribution.

10 One can refer to Dhaene et al. (2002) for some detailed discussions of comonotonic risk aggregation in actuarial practice.